No. The first statement is always true, the second, plus case law, determines the validity of the exceptions. For example, a support group for people with cervical cancer can clearly exclude trans women.
I am reasonably well versed in the case law around this, with Mackereth being the most recent. Certainly all the cases referred to in the article, plus FPW and Sex Matters. If you know of others, I'll take a look.
The most interesting will be the attempt to force trans exclusion on the Brighton Rape Crisis Centre. The law says that a provider CAN exclude trans women, and I can se no issue under the law with a RCC deciding to set up a trans-exclusionary centre for gender critical women (Rowling has).
This case will determine if they can be forced to do so. It's quite a clever framing, because it doesn't demand that trans women are excluded, merely that there be a trans-exclusive group. It's a wedge. I look forward to seeing the judgment.